Step | Hyp | Ref
| Expression |
1 | | lsmelvalm.t |
. . 3
⊢ (𝜑 → 𝑇 ∈ (SubGrp‘𝐺)) |
2 | | lsmelvalm.u |
. . 3
⊢ (𝜑 → 𝑈 ∈ (SubGrp‘𝐺)) |
3 | | eqid 2738 |
. . . 4
⊢
(+g‘𝐺) = (+g‘𝐺) |
4 | | lsmelvalm.p |
. . . 4
⊢ ⊕ =
(LSSum‘𝐺) |
5 | 3, 4 | lsmelval 19063 |
. . 3
⊢ ((𝑇 ∈ (SubGrp‘𝐺) ∧ 𝑈 ∈ (SubGrp‘𝐺)) → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) |
6 | 1, 2, 5 | syl2anc 587 |
. 2
⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) |
7 | 2 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑈 ∈ (SubGrp‘𝐺)) |
8 | | eqid 2738 |
. . . . . . . . 9
⊢
(invg‘𝐺) = (invg‘𝐺) |
9 | 8 | subginvcl 18577 |
. . . . . . . 8
⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ 𝑈) |
10 | 7, 9 | sylan 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → ((invg‘𝐺)‘𝑥) ∈ 𝑈) |
11 | | eqid 2738 |
. . . . . . . . 9
⊢
(Base‘𝐺) =
(Base‘𝐺) |
12 | | lsmelvalm.m |
. . . . . . . . 9
⊢ − =
(-g‘𝐺) |
13 | | subgrcl 18573 |
. . . . . . . . . . 11
⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
14 | 1, 13 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 ∈ Grp) |
15 | 14 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝐺 ∈ Grp) |
16 | 11 | subgss 18569 |
. . . . . . . . . . . 12
⊢ (𝑇 ∈ (SubGrp‘𝐺) → 𝑇 ⊆ (Base‘𝐺)) |
17 | 1, 16 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑇 ⊆ (Base‘𝐺)) |
18 | 17 | sselda 3916 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑦 ∈ (Base‘𝐺)) |
19 | 18 | adantr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝑦 ∈ (Base‘𝐺)) |
20 | 11 | subgss 18569 |
. . . . . . . . . . 11
⊢ (𝑈 ∈ (SubGrp‘𝐺) → 𝑈 ⊆ (Base‘𝐺)) |
21 | 7, 20 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → 𝑈 ⊆ (Base‘𝐺)) |
22 | 21 | sselda 3916 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝐺)) |
23 | 11, 3, 12, 8, 15, 19, 22 | grpsubinv 18461 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑦 −
((invg‘𝐺)‘𝑥)) = (𝑦(+g‘𝐺)𝑥)) |
24 | 23 | eqcomd 2744 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑦(+g‘𝐺)𝑥) = (𝑦 −
((invg‘𝐺)‘𝑥))) |
25 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑧 = ((invg‘𝐺)‘𝑥) → (𝑦 − 𝑧) = (𝑦 −
((invg‘𝐺)‘𝑥))) |
26 | 25 | rspceeqv 3565 |
. . . . . . 7
⊢
((((invg‘𝐺)‘𝑥) ∈ 𝑈 ∧ (𝑦(+g‘𝐺)𝑥) = (𝑦 −
((invg‘𝐺)‘𝑥))) → ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧)) |
27 | 10, 24, 26 | syl2anc 587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧)) |
28 | | eqeq1 2742 |
. . . . . . 7
⊢ (𝑋 = (𝑦(+g‘𝐺)𝑥) → (𝑋 = (𝑦 − 𝑧) ↔ (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧))) |
29 | 28 | rexbidv 3224 |
. . . . . 6
⊢ (𝑋 = (𝑦(+g‘𝐺)𝑥) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧) ↔ ∃𝑧 ∈ 𝑈 (𝑦(+g‘𝐺)𝑥) = (𝑦 − 𝑧))) |
30 | 27, 29 | syl5ibrcom 250 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑥 ∈ 𝑈) → (𝑋 = (𝑦(+g‘𝐺)𝑥) → ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |
31 | 30 | rexlimdva 3211 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) → ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |
32 | 8 | subginvcl 18577 |
. . . . . . . 8
⊢ ((𝑈 ∈ (SubGrp‘𝐺) ∧ 𝑧 ∈ 𝑈) → ((invg‘𝐺)‘𝑧) ∈ 𝑈) |
33 | 7, 32 | sylan 583 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → ((invg‘𝐺)‘𝑧) ∈ 𝑈) |
34 | 18 | adantr 484 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → 𝑦 ∈ (Base‘𝐺)) |
35 | 21 | sselda 3916 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → 𝑧 ∈ (Base‘𝐺)) |
36 | 11, 3, 8, 12 | grpsubval 18438 |
. . . . . . . 8
⊢ ((𝑦 ∈ (Base‘𝐺) ∧ 𝑧 ∈ (Base‘𝐺)) → (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) |
37 | 34, 35, 36 | syl2anc 587 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) |
38 | | oveq2 7240 |
. . . . . . . 8
⊢ (𝑥 = ((invg‘𝐺)‘𝑧) → (𝑦(+g‘𝐺)𝑥) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) |
39 | 38 | rspceeqv 3565 |
. . . . . . 7
⊢
((((invg‘𝐺)‘𝑧) ∈ 𝑈 ∧ (𝑦 − 𝑧) = (𝑦(+g‘𝐺)((invg‘𝐺)‘𝑧))) → ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥)) |
40 | 33, 37, 39 | syl2anc 587 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥)) |
41 | | eqeq1 2742 |
. . . . . . 7
⊢ (𝑋 = (𝑦 − 𝑧) → (𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥))) |
42 | 41 | rexbidv 3224 |
. . . . . 6
⊢ (𝑋 = (𝑦 − 𝑧) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑥 ∈ 𝑈 (𝑦 − 𝑧) = (𝑦(+g‘𝐺)𝑥))) |
43 | 40, 42 | syl5ibrcom 250 |
. . . . 5
⊢ (((𝜑 ∧ 𝑦 ∈ 𝑇) ∧ 𝑧 ∈ 𝑈) → (𝑋 = (𝑦 − 𝑧) → ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) |
44 | 43 | rexlimdva 3211 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧) → ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥))) |
45 | 31, 44 | impbid 215 |
. . 3
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑇) → (∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |
46 | 45 | rexbidva 3223 |
. 2
⊢ (𝜑 → (∃𝑦 ∈ 𝑇 ∃𝑥 ∈ 𝑈 𝑋 = (𝑦(+g‘𝐺)𝑥) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |
47 | 6, 46 | bitrd 282 |
1
⊢ (𝜑 → (𝑋 ∈ (𝑇 ⊕ 𝑈) ↔ ∃𝑦 ∈ 𝑇 ∃𝑧 ∈ 𝑈 𝑋 = (𝑦 − 𝑧))) |